Examples of special sphere bundles I'm interested in examples of sphere bundles which do not arise from vector bundles.
I'm not quite clear about the following. So please let me know if anything is false.
I believe that a $(k-1)$-sphere bundle arises from a vector bundle iff its structure group can be reduced to $O(k)$. I think that for $k\geq 4$ it is not known if $O(k)$ is homotopy equivalent to $\operatorname{Diff}(S^{k-1})$, and in general it is false. So there are sphere bundles that do not arise from vector bundles.
I'm also interested in more details/clarification of this argument.
 A: Your argument starts on the right track but you haven't said enough. A smooth $S^{k-1}$-bundle on a space $X$ is classified by a map $X \to B \text{Diff}(S^{k-1})$. The natural action of $O(k)$ on $S^{k-1}$ induces a map $BO(k) \to B \text{Diff}(S^{k-1})$, and the question is whether a map into $B \text{Diff}(S^{k-1})$ always admits a lift (up to homotopy) to a map into $BO(k)$.
The reason you haven't said enough is that this can be possible without the map $BO(k) \to B \text{Diff}(S^{k-1})$ being an equivalence. The lifting problem is always solvable iff it's solvable for the universal example, namely the identity map $B \text{Diff}(S^{k-1}) \to B \text{Diff}(S^{k-1})$. A lift of this map into $BO(k)$ is precisely a homotopy section of the natural map $BO(k) \to B \text{Diff}(S^{k-1})$ (that is, a section, up to homotopy). In particular, you don't need a homotopy inverse (as I think is being claimed in the comments), only a homotopy right inverse.
If a map between spaces has a homotopy right inverse then applying any homotopy-invariant functor produces a map with a right inverse, hence in particular a map which is surjective. So to rule this possibility out it suffices to show that the induced map on, say, $H_n$ for some $k$ fails to be surjective, or that the induced map on $H^n$ for some $k$ fails to be injective. But I don't know enough about $B \text{Diff}(S^{k-1})$ to do this. 
